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Introduction to Quantum Groups I Introduction to Quantum Groups I. Groups, Algebras and Duality Pierre Clare William & Mary Sept. 30, 2020 1 / 19 Announcements Future talks Next on Introduction to Quantum Groups • Oct. 7: q-Deformations of Lie algebras E. Shelburne’21 • Oct. 21: Quantum automorphisms of graphs S. Phillips’21 and A. Pisharody’21 • Oct. 28: TBD M. Weber (Saarland University) • Nov. 11: TBD E. Swartz 2 / 19 Announcements Takeaways of this talk Shocking Revelation Noncommutative Geometry is pointless and quantum groups don’t really exist. Oversimplification Quantum objects are given in terms of algebraic structures obtained by altering structures associated with classical objects. Outline of future talks Lie group Lie algebra Enveloping algebra Quantum group q-deformation G { g { U(g) { Uq(g) Graph Automorphisms C*-algebra Quantum automorphisms liberation Γ { Aut(Γ) { C(Aut(Γ)) { C(Aut+(Γ)) 3 / 19 Announcements Takeaways of this talk Today’s objectives 1 C*-algebras are the right language for quantum (noncommutative) spaces. 2 Generic instance of liberation: quantum groups à la Woronowicz Compact group Commutative C*-algebra Woronowicz algebra G { C(G) { A (Matrix quantum group) 4 / 19 Noncommutative spaces From spaces to algebras Let X be a locally compact (Hausdorff) space. No algebraic structure. Consider: n o C0(X) = f : X −! C ; continuous with lim f(x) = 0 : x!1 Then C0(X) is an algebra over C: (λf + g)(x) = λf(x) + g(x) ; (fg)(x) = f(x)g(x): It also carries a norm: n o kfk = sup jf(x)j ; x 2 X for which it is complete, and an involution f ∗(x) = f(x) such that kf ∗fk = kfk2: In other words, C0(X) is a C*-algebra. Remark: multiplication in C0(X) is commutative. 5 / 19 Noncommutative spaces Why look at C0(X) instead of X? Topological properties of X can be read off C0(X). For instance: Theorem X is compact , C0(X) has a multiplicative identity. Proof: a multiplicative identity e of C0(X) must satisfy e(x)f(x) = f(x) for all x 2 X and every f 2 C0(X). Necessarily, e(x) = 1 for all x 2 X. Constant functions are continuous and lim 1 = 0 if and only if X is compact x!1 Points in X give continuous linear functionals on C0(X). For x 2 X, consider eval : C (X) −! C x 0 : f 7−! f(x) 6 / 19 Noncommutative spaces Topological spaces = commutative C*-algebras Every (locally compact) space X gives rise to a C*-algebra C0(X). What C*-algebras are obtained in this way? Obviously, only commutative C*-algebras. All of them? Theorem (Gelfand-Naimark-Segal, 1943) EveryC*-algebra A is isomorphic to aClosed ∗-invariant subalgebra of continuous linear operators on some Hilbert space H: A ⊂ B(H): Theorem (Gelfand-Naimark, 1943) If A is a commutativeC ∗-algebra, there exists a locally compact Hausdorff space X (the spectrum of A) such that A ' C0(X): Classical spaces ! Commutative C*-algebras. 7 / 19 Noncommutative spaces Example: the noncommutative torus Let θ 2 R. Consider a unital C*-algebra Aθ generated by two elements U and V such that U∗U = 1 ; V ∗V = 1 ; UV = e2iπθVU: If θ = 0 then Aθ is commutative, hence an algebra of functions by Gelfand-Naimark’s theorem. In fact, 2 A0 ' C(T ) where T2 = S1 × S1 is the 2-torus. In general, Aθ is not commutative, so not an algebra of functions... but we still write 2 Aθ = C(Tθ ) 2 and say that Tθ is a noncommutative torus. Just because a space doesn’t exist, doesn’t mean it can’t have nice properties. 2 Theorem. Tθ is compact. Proof (interpretation): Aθ contains a multiplicative unit. 2 Remark: Tθ may not have points but Aθ has linear functionals, representations... 8 / 19 Towards quantum groups General strategy Gelfand theory says: ∼ Classical spaces ! Commutative C*-algebras X 7−! C0(X) Relaxing the commutativity condition, Quantum spaces ! (Possibly) noncommutative C*-algebras 7−! A What is a quantum group? Strategy: ∗ 1 Associate algebras to ordinary groups. k[G], Cr (G), L(G), C(G), g, U(g)... 2 Characterize them. 3 Relax conditions to get new objects (and call them quantum groups). What algebras should we choose? 9 / 19 Pontryagin duality Duality and Fourier transform Let G be a locally compact abelian group: Zn, Z, T, R, Qp , ... The Pontryagin dual of G is n o Gb = Hom(G; T) = χ : G −! C ; cont. hom. with jχ(g)j ≡ 1 Theorem (Pontryagin Duality) Gb is a locally compact abelian group and Gb ' G. Lev Pontryagin 1908 - 1988 Examples of characters of T and R: n ixξ χn(z) = z and χξ(x) = e n o • bR = χξ ; ξ 2 R ' R n o • bT = χn ; n 2 Z ' Z and by Pontryagin duality, bZ ' T. Theorem The Pontryagin dual of a compact (resp. discrete) abelian group is discrete (resp. compact.) Now, back to the main question... 10 / 19 Pontryagin duality Duality and Fourier transform What algebra related to a group G should we study (and then deform)? • C0(G): functions on G with pointwise multiplication • C*-algebra, always commutative, • unital if and only if G is compact (then C0(G) = C(G)). ∗ • Cr (G): (completion of) Cc (G) with convolution (= C[G] if G is finite) • C*-algebra, commutative if and only if G is abelian, • unital if and only if G is discrete. Theorem (Fourier Analysis on Abelian Groups) Let G be an abelian group. For f 2 Cc (G) and χ 2 G,b let Z bf(χ) = f(g) χ(g) dg: G Thenbf 2 C0(Gb) and the Fourier transformation f 7!bf extends to an isomorphism ∗ Cr (G) ' C0(Gb): ∗ If G is abelian, C0(G) and Cr (G) are dual choices. If G is non-abelian and non-compact, Gb is not a group... 11 / 19 Matrix quantum groups Characterization of C(G) Goal for the rest of the talk ∗ Describe a framework that accommodates algebras such as C0(G) and Cr (G), in which Fourier-Pontryagin duality is restored. Neither G nor Gb will really exist, but they will be dual to each other. ::: c First, let us characterize C0(G) when G is a classical group. Two questions • How does the algebraic structure of G manifest at the level of C0(G)? • Can C0(G) be described by generators and relations? 12 / 19 Matrix quantum groups Characterization of C(G) Let G be a group with identity e. The algebra C(G) carries: ◦ a comultiplication ∆ : C(G) −! C(G × G): ∆(f):(g1; g2) 7−! f(g1g2) ◦ a counit " : C(G) −! C: "(f) = f(e) ◦ an antipode S : C(G) −! C(G): S(f): g 7−! f(g−1) Since G is a group, ∆, " and S satisfy: • Coassociativity: (∆ ⊗ id) ◦ ∆ = (id ⊗ ∆) ◦ ∆ f((g1g2)g3) = f(g1(g2g3)) • Counitality: (" ⊗ id) ◦ ∆ = id = (id ⊗ ") ◦ ∆ f(e ·) = f = f(· e) • Coinvertibility: m ◦ (S ⊗ id) ◦ ∆ = η ◦ " = m ◦ (id ⊗S) ◦ ∆ f(g−1g) = f(e) = f(gg−1) The algebraic structure of G is encoded in C(G); ∆; "; η, S . 13 / 19 Matrix quantum groups Characterization of C(G) Now assume that G is a compact (Lie) group (of unitary matrices): G ⊂ U(n): Consider the elementary matrix coefficient functions uij 2 C(G): 2 g ::: g 3 6 11 1n 7 6 : 7 6 :: 7 6 7 6 : : 7 uij : 6 : g : 7 7−! gij 6 ij 7 6 : 7 6 :: 7 46 57 gn1 ::: gnn Facts: • The functions fuij ; 1 ≤ i; j ≤ ng generate the C*-algebra C(G). • They form a biunitary matrix u = [uij ] in Mn(C(G)). • P ∆(uij ) = k uik ⊗ ukj . It is a morphism ∆ : C(G) −! C(G) ⊗ C(G). • "(uij ) = δij . It is a morphism " : C(G) −! C. • ∗ op S(uij ) = uji = uji defines a morphism S : C(G) −! C(G) . 14 / 19 Matrix quantum groups Liberation of C(G) Woronowicz algebras Characterization of C(G) Definition (Woronowicz, 1987) A matrix quantum group is a unital C*-algebra A with maps ∆, " and S such that: 2 • A is generated by n elements fuij ; 1 ≤ i; j ≤ ng. • They form a biunitary matrix u = [uij ] in Mn(A). • P ∆(uij ) = k uik ⊗ ukj defines a morphism ∆ : A −!A⊗A . • "(uij ) = δij defines a morphism " : A −! C. • ∗ op S(uij ) = uji defines a morphism S : A −!A . 15 / 19 Matrix quantum groups Characterization of compact groups Recall the correspondence given by Gelfand theory: ∼ Locally compact spaces ! Commutative C*-algebras X 7−! C0(X) Now we have: Compact groups −! Commutative Wornowicz algebras G 7−! C(G) { Is this a one-to-one correspondence? Theorem Every commutative Woronowicz algebra is of the form C(G), where G is a compact group. Proof: let A be a commutative Woronowicz algebra. Since A is unital, it has compact spectrum. Let G = Sp(A) and consider the embedding Sp(A) 3 ' 7−! ['(uij )] 2 U(n): Properties of ∆, " and S show that the image of this embedding is a subgroup of U(n). 16 / 19 Matrix quantum groups Another example Are there interesting examples of noncommutative Woronowicz algebras? Let Γ = hγ1; : : : ; γni be a finitely generated group and consider the associated C*-algebra: dense C∗(Γ) ⊃ C[Γ] ⊃ Γ: It is a Woronowicz algebra with defining matrix: 2 3 6 γ1 7 6 7 6 γ2 7 u = 6 7 2 M (C∗(Γ)) 6 :: 7 n 6 : 7 46 57 γn and structure maps defined by: ∆(γ) = γ ⊗ γ ; "(γ) = 1 ; S(γ) = γ−1: Theorem All co-commutative Woronowicz algebras are of the form C∗(Γ). 17 / 19 Matrix quantum groups Pontryagin duality revisited General picture Let A be a Woronowicz algebra.
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